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Results tagged with lie-algebras
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user 138296
Lie algebras are algebraic structures which were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used. Related mathematical concepts include Lie groups and differentiable manifolds.
2
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An inequality for weights of affine Lie algebras, level, and dual Coxeter number
Suppose $\mathfrak{g}$ is an (untwisted) affine Lie algebra with the normalized invariant form $(\cdot | \cdot)$. Let $\lambda \in \mathfrak{h}^\ast$ be a dominant integral weight such that $\lambda(d …
- 550
5
votes
0
answers
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Motivation and Difference of Category O Definition for Kac-Moody Algebras
My first encounter with Category $\mathcal{O}$ was (perhaps unusually) learning about Kac-Moody algebras from Kac's book. Kac takes the following definition:
The Category $\mathcal{O}$ has objects $\ …
- 550
4
votes
Classification of root lattice embeddings in $E_{10}$
The results I am familiar with in this direction are in the related area of regular subalgebras, or $\pi$-systems, as in Carbone et. al. here. There, they give a good amount of information about the p …
- 550
7
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0
answers
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Demazure modules and dimension of weight spaces
Let $\mathfrak{g}$ be a symmetrizable Kac–Moody algebra, $w \in W$ an element of the Weyl group, and $\lambda$ an integral dominant weight with $V(\lambda)$ the associated irreducible highest weight r …
- 550
1
vote
Accepted
Action of the Casimir on highest weight modules for Kac-Moody algebra
You should be a bit careful, as this isn't precisely the action of the Casimir on $v \otimes v$, but instead follows from it.
For each positive root $\alpha$, let $e_\alpha^{(1)}, \dots, e_\alpha^{(n_ …
- 550
5
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0
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Weyl Group Action on Littelmann Paths
In his paper "Paths and Root Operators in Representation Theory," Littelmann gives an action of the Weyl group on the set of integral paths via
$$
\tilde{s}_\alpha(\pi):= \begin{cases} f^n_\alpha(\pi …
- 550
3
votes
0
answers
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Multiplicity relation between highest weight modules, Demazure modules, and crystals
Let $\mathfrak{g}$ be a symmetrizable Kac--Moody algebra, and let $\lambda$ be an associated dominant integral weight. Then two different objects we can relate to this data is $V(\lambda)$, the irredu …
- 550
1
vote
Source for highest weight vectors for $\text{SL}_n(\mathbb{C})$ representations
If you are trying to recognize the irreducible representations in some larger ambient representation, then the form of the highest weight vector(s) will depend heavily on that scenario. So your questi …
- 550
3
votes
Accepted
Difference between two definitions of affine Lie algebras
In my experience, the Laurent polynomial construction is more suited to the "algebraic" aspects of the theory--in particular, if the power of $t$ corresponds to the coefficient of $\delta$ in the root …
- 550
6
votes
Sum of two positive roots which is not a root: uniqueness of heights of the summands
Here's a naive approach which I think works in the simply-laced cases, unless I've made a silly mistake somewhere. The restriction to simply-laced cases allows me to play fast and loose with the root- …
- 550